After connecting the sounder it sounds, but if you disconnect the vehicle battery will it stop, this means that the sounder battery is faulty but that will not affect the system.
In theory if the sounder has a fully charged battery you could be unplug, unbolted and removed from the vehicle and walk down the road with it still sounding.
It's possible that the alarm ECU is in an alarm mode and needs to be reset or has locked up in the alarm mode or may be faulty and they do go faulty as they are powered up 24/7.
I don't know how you can tell that the alarm is working other than looking at the indicators flash, then setting it off, but then you need the sounder connected.